Three contributions to topology, algebraic geometry and representation theory: Homological finiteness of abelian covers, algebraic elliptic cohomology theory and monodromy theorems in the elliptic setting.

摘要

My doctoral work consists of three projects. The first project is joint work with A. Suciu and G. Zhao described in more details in Chapter 2 and 3 of this dissertation. In Chapter 2, we exploit the classical correspondence between finitely generated abelian groups and abelian complex algebraic reductive groups to study the intersection theory of translated subgroups in an abelian complex algebraic reductive group, with special emphasis on intersections of (torsion) translated subtori in an algebraic torus. In Chapter 3, we present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CW-complex X, with group of deck transformations a fixed abelian group A, which in the case of free abelian covers of rank r coincides with the Grassmanian of r-planes in H1( X, Q ). Inside this parameter space, there is a subset O Ai(X) consisting of all the covers with finite Betti numbers up to degree i. Building on work of Dwyer and Fried, we show how to compute these sets in terms of the jump loci for homology with coefficients in rank 1 local systems on X. For certain spaces, such as smooth, quasi-projective varieties, the generalized Dwyer--Fried invariants that we introduce here can be computed in terms of intersections of algebraic subtori in the character group. For many spaces of interest, the homological finiteness of abelian covers can be tested through the corresponding free abelian covers. Yet in general, abelian covers exhibit different homological finiteness properties than their free abelian counterparts. The second project is joint work with M. Levine and G. Zhao described in more details in Chapter 4 of this dissertation. We define the algebraic elliptic cohomology theory coming from Krichever's elliptic genus as an oriented cohomology theory on smooth varieties over an arbitrary perfect field. We show that in the algebraic cobordism ring with rational coefficients, the ideal generated by differences of classical flops coincides with the kernel of Krichever's elliptic genus. This generalizes a theorem of B. Totaro in the complex analytic setting. The third project consists of two parts. Chapter 5 is joint work with N. Guay, where we give a double loop presentation of the deformed double current algebras, which are deformations of the central extension of the double current algebras g [u, v], for a simple Lie algebra g g. We prove some nice properties of the algebras using the double loop presentation. Especially, we construct a central element of the deformed double current algebra. Chapter 6 is joint work with V. Toledano Laredo. Calaque-Enriquez Etingof constructed the universal KZB equation, which is a flat connection on the configuration space of n points on an elliptic curve. They show that its monodromy yields an isomorphism between the completions of the group algebra of the elliptic braid group of type An-1 and the holonomy algebra of coefficients of the KZB connection. We generalized this connection and the corresponding formality result to an arbitrary root system. We also gave two concrete incarnations of the connection: one valued in the rational Cherednik algebra of the corresponding Weyl group, the other in the double deformed current algebra D( g ) of the corresponding Lie algebra g . The latter is a deformation of the double current algebra g [u,v] recently defined by Guay, and gives rise to an elliptic version of the Casimir connection.